The talk will illustrate two themes. The first is that topology, a branch of mathematics that was developed during the 20th century, can be viewed in part as a natural continuation of mathematicians' efforts, from the 17th century onward, to clarify the foundations of calculus. The second is that, in contrast to sciences in which a new theory supplants all previous ones, in mathematics new developments generally arise as extensions of previous discoveries. Specifically, we will trace a portion of the history of calculus and topology to see how it served as background to research conducted late in the 20th century.